Yes, mostly.
There exist items that do pretty much the same things, at least as some of the properties, but these are described as specific items. For instance, I think you'll find a flaming sword in the DMG, but no flaming mace, axe, or bow.
Some items specifically say what kind of weapon they are (sunblade is a long sword, for example) while others (Vicious weapon) can be any type as determined by the DM.
However, the WoTC team has said in SA that you can reskin just about anything, as the DM. If you want a flametongue mace, go for it. One of my players has a crossbow that fires radiant arrows that it makes (like the bow in the old D&D cartoon), while another has a spear that does lightning damage.
Also, there are four tables in the DMG that allow you to customize each magic item based on creator or intended use, historical detail, minor property, and quirk. That hearkens back to the BECMI tables for weapon generation, and you can certainly use those in your 5e game with very little modification. Here's my conversions for melee weapons:
Melee Weapon Talents
Talents are unusual magical powers that various weapons can have (though they are most often applied to swords). They are not the same as a sword's primary or extraordinary powers, which are described later, and a sword can have both a talent and primary and extraordinary powers. Talents can be certain spell effects or some other useful ability.
All talents may be used only once per long rest unless noted otherwise. Talents that replicate spells do not require preparation or spell slots, and they don’t require somatic or material components or concentration. Unless otherwise noted, using a talent requires a bonus action to activate.
·
Breathing. The weapon with this talent allows the user to cast
water breathing (self only) as an action, once per long rest. If used in a non-liquid environment, it supplies only the user with pure air for 10 minutes, and it can be used to counter the effects of airlessness, poisoned air (such as a gas trap), and so forth; however, it cannot negate the effects of any breath weapon.
·
Charming. The weapon with this talent allows the user to cast
charm person as an action, once per short rest.
·
Deceiving. The weapon with this talent allows the user to cast
disguise self as an action, once per short rest.
·
Deflecting. When another creature damages the wielder with a melee attack, the wielder can use their reaction to reduce the damage by 1d10 + their Dexterity modifier. This ability can be used once per short rest.
·
Finding. The weapon with this talent allows the user to cast one
locate object or
locate plant or animal spell as an action, twice per long rest. The very rare version can cast
locate creature spell as an action, once per long rest.
·
Healing. The weapon with this talent comes in two different versions. One allows the user to cast
prayer of healing as an action, once per long rest. The effect heals 3d8+3 hit points for each creature selected. The other allows the user to cast
cure wounds as an action, once per long rest. The effect heals 4d8+4 hit points.
·
Hiding. The weapon with this talent allows the user to cast
invisibility (self only) as an action, once per long rest. In addition, the weapon can itself become invisible three times per long rest (with the same restrictions as the
invisibility spell).
·
Holding. The weapon with this talent allows the user to cast
hold person as an action, once per long rest.
·
Lighting. The weapon with this talent allows the user to cast
light as a bonus action, up to three times per long rest.
·
Silencing. The weapon with this talent allows the user to cast
silence as an action, once per long rest.
·
Slowing. When a successful hit is made, the weapon with this talent allows the user to cast
slow as a bonus action. The spell only targets the creature struck. The user may decide whether or not to use this effect after the swing hits. This talent can only be used once per long rest.
·
Speeding. The weapon with this talent allows the user to cast
haste (on the user only) as an action, once per long rest.
·
Translating. The weapon with this talent allows the user to cast
comprehend languages as a bonus action, up to three times per long rest.